Calculate the impulse of the force N from time t = 0 s to time

t = 2 s.

Solution:

EXAMPLE 3: THE LINEAR IMPULSE-MOMENTUM RELATION

A projectile of mass m=10 kg having an initial velocity of m/s is

subjected to a horizontal wind force as shown in the figure. Calculate the

projectiles velocity just after the wind has ended.

Solution:

Linear impulse-momentum relation:

Substitution into (1) gives:

EXAMPLE 4: THE LINEAR IMPULSE-MOMENTUM RELATION

The two particles are moving at velocity and just

before they collide and become connected. Calculate the velocity of the

system after the collision.

Solution:

Substitution into (1) gives

Work Energy Principle

EXAMPLE 1: THE WORK-ENERGY RELATION

Calculate the power of the frictional force to do work on the grinding disk

and its power to do work on the work piece B. Assume the grinding wheel

and connection have a total mass of m. What happens to the difference in the power?

Solution:

(a) Power of friction to do work on the grinding wheel

Kinetics:

Kinematics:

(b) Power of friction to do work on work piece B

The difference in the power between PA and PB appears in the form of heat that ends up heating the surfaces.

EXAMPLE 2: THE WORK-ENERGY RELAITON

Calculate the work from time 0 to time t of the constant magnitude force

that changes its direction by . Assume A moves with constant speed

.

Solution:

EXAMPLE 3: THE WORK-ENERGY RELATION

A particle moves under the constant horizontal force and gravity from

point A to B on the smooth circular bar. Calculate the work of the forces

applied on the particle.

Solution:

Work of N: N is perpendicular to the path at all times and, as a result, has no component tangent to the path . Therefore, N does no work from A to

B.

Work of and mg:

EXAMPLE 4: THE WORK-ENERGY RELATION

The particle starts from rest at A and moves along the smooth circular bar

under the constant horizontal force and under gravity. Calculate the

velocity of the particle as a function of .

Solution:

Work of N equals zero since it is perpendicular to the path at all the times.

Work of and mg

Work-Energy relation:

start from rest)

(1) into (2)

EXAMPLE 5: THE WORK-ENERGY RELATION

The particle starts from rest at A. If the bar is smooth and the spring has a

free length of , calculate the speed of the particle as a function of x.

Solution:

First calculate the work of the forces.

N does no work since it is perpendicular to the path of the bar at all

times.

Start from rest

Work-Energy relation

EXAMPLE 1: THE ANGULAR

IMPULSE-MOMENTUM RELATION

A particle of mass m=10 kg is at position (1,2,3) and has a velocity

m/s. Calculate the angular momentum of this particle about

point O with coordinates (0,2,1). The coordinates are given in meters.

Solution:

EXAMPLE 2: THE ANGULAR IMPULSE-MOMENTUM RELATION

A particle of mass m=5 kg is at the position shown and has a velocity

m/s. Calculate the angular momentum of this particle about

point P.

Solution:

Assuming counter clockwise to be positive

EXAMPLE 3: THE ANGULAR IMPULSE-MOMENTUM RELATION

The motor M applies a constant moment to rotate the particle. If the

particle of mass m=20 kg starts from rest and the motor applies a moment

=10 N-m, calculate the speed of the particle after 10 seconds. Assume

the particle is fixed to the bar at a radius of 0.1 m.

Solution:

substitution into (1) gives

EXAMPLE 4: THE ANGULAR IMPULSE-MOMENTUM RELATION

Particle A of mass 1 kg moves on a smooth horizontal surface and is

connected by a string to particle B. B has a mass of 2 kg. Particle B is

released from rest at time t=0 when particle A has a circumferential velocity of 10 m/s and a radius of 0.5 m. Write the equation of motion of the system.

Solution:

Particle A:

Particle B:

Kinematics relation:

Add (3) and (4) and substitute (5) to get

Substitution from (2) gives

Equation (2) and (6) are the equation of motion for the system.